1. On non-normal arc-transitive 4-valent dihedrantsIstván Kovács, Boštjan Kuzman, Aleksander Malnič, 2010, original scientific article Abstract: Let ▫$X$▫ be a connected non-normal 4-valent arc-transitive Cayley graph on a dihedral group ▫$D_n$▫ such that ▫$X$▫ is bipartite, with the two bipartition sets being the two orbits of the cyclic subgroup within ▫$D_n$▫. It is shown that ▫$X$▫ is isomorphic either to the lexicographic product ▫$C_n[2K_1]$▫ with ▫$n \geq 4$▫ even, or to one of the five sporadic graphs on 10, 14, 26, 28 and 30 vertices, respectively. Found in: ključnih besedah Summary of found: ...non-normal 4-valent arc-transitive Cayley graph on a dihedral group ▫$D_n$▫ such that ▫$X$▫ is bipartite,... Keywords: Cayley graph, arc transitivity, dihedral group Published: 15.10.2013; Views: 1800; Downloads: 42 Full text (0,00 KB) |
2. Classification of 2-arc-transitive dihedrantsShao Fei Du, Aleksander Malnič, Dragan Marušič, 2008, original scientific article Abstract: A complete classification of 2-arc-transitive dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing the study of these graphs initiated by the third author in [D. Marušič, On 2-arc-transitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003) 162-196]. The list consists of the following graphs: (i) cycles ▫$C_{2n},\; n \ge 3$▫; (ii) complete graphs ▫$K_{2n}, \; n \ge 3$▫; (iii) complete bipartite graphs ▫$K_{n,n}, \; n \ge 3$▫; (iv) complete bipartite graphs minus a matching ▫$K_{n,n} - nK_2, \; n \ge 3$▫; (v) incidence and nonincidence graphs ▫$B(H_{11})$▫ and ▫$B'(H_{11})$▫ of the Hadamard design on 11 points; (vi) incidence and nonincidence graphs ▫$B(PG(d,q))$▫ and ▫$B'(PG(d,q))$▫, with ▫$d \ge 2$▫ and ▫$q$▫ a prime power, of projective spaces; (vii) and an infinite family of regular ▫${\mathbb{Z}}_d$▫-covers ▫$K_{q+1}^{2d}$▫ of ▫$K_{q+1, q+1} - (q+1)K_2$▫, where ▫$q \ge 3$▫ is an odd prime power and ▫$d$▫ is a divisor of ▫$\frac{q-1}{2}$▫ and ▫$q-1$▫, respectively, depending on whether ▫$q \equiv 1 \pmod{4}$▫ or ▫$q \equiv 3 \pmod{4}$▫ obtained by identifying the vertex set of the base graph with two copies of the projective line ▫$PG(1,q)$▫, where the missing matching consists of all pairs of the form ▫$[i,i']$▫, ▫$i \in PG(1,q)$▫, and the edge ▫$[i,j']$▫ carries trivial voltage if ▫$i=\infty$▫ or ▫$j=\infty$▫, and carries voltage ▫$\bar{h} \in {\mathbb{Z}}_d$▫, the residue class of ▫$h \in {\mathbb{Z}}_d$▫, if and only if ▫$i-j = \theta^h$▫, where ▫$\theta$▫ generates the multiplicative group ▫${\mathbb{F}}_q^\ast$▫ of the Galois field ▫${\mathbb{F}}_q$▫. Found in: ključnih besedah Summary of found: ...2-arc-transitive dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing the study... Keywords: permutation group, imprimitive group, dihedral group, Cayley graph, dihedrant, 2-Arc-transitive graph Published: 15.10.2013; Views: 1606; Downloads: 50 Full text (0,00 KB) |
3. Distance-regular Cayley graphs on dihedral groupsŠtefko Miklavič, Primož Potočnik, 2007, original scientific article Abstract: The main result of this article is a classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such graphs, which are called trivial. These are: complete graphs, complete bipartite graphs, complete bipartite graphs with the edges of a 1-factor removed, and cycles. It is proved that every non-trivial distance-regular Cayley graph on a dihedral group is bipartite, non-antipodal, has diameter 3 and arises either from a cyclic di#erence set, or possibly (if any such exists) from a dihedral difference set satisfying some additional conditions. Finally, all distance-transitive Cayley graphs on dihedral groups are determined. It transpires that a Cayley graph on a dihedral group is distance-transitive if and only if it is trivial, or isomorphic to the incidence or to the non-incidence graph of a projective space ▫$\mathrm{PG}_{d-1} (d,q)$▫, ▫$d \ge 2$▫, or the unique pair of complementary symmetric designs on 11 vertices. Found in: ključnih besedah Summary of found: ...classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such... Keywords: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference set Published: 15.10.2013; Views: 1276; Downloads: 57 Full text (0,00 KB) |
4. Distance-regular Cayley graphs on dihedral groupsPrimož Potočnik, Štefko Miklavič, 2005, original scientific article Abstract: The main result of this article is a classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such graphs, which are called trivial. These are: complete graphs, complete bipartite graphs, complete bipartite graphs with the edges of a 1-factor removed, and cycles. It is proved that every non-trivial distance-regular Cayley graph on a dihedral group is bipartite, non-antipodal, has diameter 3 and arises either from a cyclic di#erence set, or possibly (if any such exists) from a dihedral difference set satisfying some additional conditions. Finally, all distance-transitive Cayley graphs on dihedral groups are determined. It transpires that a Cayley graph on a dihedral group is distance-transitive if and only if it is trivial, or isomorphic to the incidence or to the non-incidence graph of a projective space ▫$\mathrm{PG}_{d-1} (d,q)$▫, ▫$d \ge 2$▫, or the unique pair of complementary symmetric designs on 11 vertices. Found in: ključnih besedah Summary of found: ...classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such... Keywords: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference set Published: 10.07.2015; Views: 880; Downloads: 47 Full text (0,00 KB) |
5. Further restrictions on the structure of finite DCI-groupsEdward Dobson, Joy Morris, Pablo Spiga, 2015, original scientific article Found in: ključnih besedah Summary of found: ...isomorphism problem, CI-group, dihedral group, Cayley graph, ... Keywords: isomorphism problem, CI-group, dihedral group, Cayley graph Published: 08.08.2016; Views: 967; Downloads: 24 Full text (0,00 KB) |
6. Which Haar graphs are Cayley graphs?István Estélyi, Tomaž Pisanski, 2016, original scientific article Found in: ključnih besedah Summary of found: ...Haar graph, Cayley graph, dihedral group, generalized dihedral group, ... Keywords: Haar graph, Cayley graph, dihedral group, generalized dihedral group Published: 16.11.2017; Views: 987; Downloads: 26 Full text (0,00 KB) |